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Technical write- up on Tidal Bulges
Some notes on tides, contributed by [email protected]
(Srinivas Bettadpur). Please send comments to him.
---
The references I have are far too technical for all this. Part A
is from my class notes, Part B can be found in
"Geophysical Geodesy" by K. Lambeck
"Tides of the planet Earth" by P. Melchior
People in Astronomy could give you much better references. Let me
know if this is too long/short (oh, yeah !). Welcome any comments.
---------------------------------------
The answers are in three parts. In the first part, a simple 2-D
case is considered as an example of mechanism of tidal deformation. In
the second, a brief mention of treatment of tides in practice is
given. In the third, an explanation of evolution of the Earth-Moon
system is given.
PART A : 2-D Example of Tidal Deformation
Since gravitational attraction is a function of the distance
between two masses, Lunar attraction on the Earth is not uniform. Some
parts of the Earth are more strongly attracted to the moon than the
others. This *differential* attraction gives rise to tides. The
reference attraction is chosen as that acting at the center of the
Earth, and the resulting *variation* from this reference is called the
tidal attraction. Note that while the gross, orbital motion of an
object is governed by the sum of forces acting at the center of mass
(CM) of an object, its deformation would be governed by difference in
forces between a reference point and the CM of the body. To see the
nature of these tidal forces and the resulting deformation, consider a
circular sheet of mass, with the moon in the same plane. If the points
on the circle at the intersection of the line joining the center C of
this circle and the moon are marked as N (for NEAR) and F (for FAR),
the forces at these three points can be drawn as
F C N ------- to Moon
:------> :--------> :---------->
(6) (8) (10)
Since the deformation of the sheet is proportional to the
*difference* in the forces at a point from those at the center, a
picture of the relative accelerations of points F, C and N can be
drawn as
F C N
<--: : :-->
(-2) (0) (+2)
This should show why, in the general case, we have tidal bulges at
both the near and far sides from the moon. This same principle can be
used to write the tidal attraction at different points along the
circumference of this circular sheet.
Y | P
| /
| /
| /
/________________ *
O X M
Ang(POX) = A , Ang(OMP) = e , OM = R , Re = Radius of circle Then,
approximately (M = Mass of Moon)
Fx = GM * Re / (R)^3 * 2 * cos (A)
Fy = GM * Re / (R)^3 * sin (A)
Draw this function from A=0 to A=360 (set rest of the multipliers
equal to one) and you will see why a circular cross section deforms
into an ellipse.
PART B : Treatment of Tidal Fields in Practice
The Earth being the messily complicated object that it is, the
picture in Part A is nowhere near adequate from practical
applications. First of all, note that a closer picture would be one
where the point M goes around O in 27.? days, whereas the axes XY
themselves spin around the the point O in 24 hours. Thus the tidal
deformation of the circle is changing in both space and time, such
that the tidal force acting on you is not the same as that on a person
in Tibet, and further, both of you will be subject to different tidal
accelerations at different times.
This spatial and temporal aspect of the variations are captured in
a position dependent function called the Tide Raising Potential (TRP),
whose spatial derivatives give the tidal accelerations at a given
point. As might be expected, this depends in a complicated way upon
the relative Earth-Moon-Sun geometry.
In the more precise work, it is usually assumed that the Earth
does not instantaneously respond to the temporal variability of the
tides. Further, the deformation in the solid earth is assumed to show
the same spatial variability as the imposed tides. That is a bad
assumption for oceans, which being much more fluid, respond in a
spatially much more intricate way than the solid Earth.
PART C : Long term evolution of the Earth-Moon system under tides
The question on this topic generally refers to the gradual
evolution of a two elastic bodies system into a state of tidal lock.
Or, in other words, these debates start with "Why does the moon
present the same face to the Earth all the time ?" There are many such
systems in the solar system, the most obvious of which is the
Pluto-Charon system. For the purposes of this discussion, I will
define tidal lock to be a situation where *BOTH* the bodies present
the same face to each other (as Earth does not, or half the people in
the world would never have seen the moon).
As mentioned in Part B, there is delay between the imposed tidal
acceleration and the Earth deformation response. If the orbital period
of the moon is different from the rotation period of the Earth, this
means that the bulge due to the deformation does not lie under the
line joining the Earth and the moon. In this case, since the rate of
rotation is larger than the rate of revolution, the bulge gets ahead
of the sub-lunar point on the Earth due to the delayed response. If
the case were reversed, the bulge would trail behind. In either case,
the phenomena is called a Tidal Lag, only the algebraic sign on the
angle is shifted depending on whether it leads or lags.
This has the net effect of causing a continuous transverse
accleration on the moon, causing it to gain velocity and raise its
orbital distance from the Earth. In reaction to delivering the kick to
the moon and rasing its orbital angular momentum, the Earth
experiences a torque that tends to slow down its rotation.
In another example, the orbital periods of Phobos and Deimos are
such in relation to Mars rotation period, that while one leads, the
other lags. Thus one experiences "drag" while the other experiences
"thrust". Thus Phobos and Deimos are said to "exchange orbital angular
momentum through the medium of Mars".
Of course, as the moon gets farther and farther away, the tidal
bulge on the Earth and consequently, the kick to the Moon will weaken.
Moreover, the deformation of the Earth in response to Lunar tides is
not without dissipation of energy (which is what causes the tidal lag
in the first place). Tidal Friction (as it is called) causes the Earth
to continually lose kinetic energy of rotation as heat, and as a
result, its rotation rate is slowing down. This is case where, if
considered in isolation from all else, the system conserves angular
momentum while losing energy.
Since the Moon is receding due to the tidal kick, its orbital
period is also slowing down. It is expected that the system will reach
equilibrium when the Moon is just far enough and the Earth just slow
enough that the tidal bulge always lies along the line joining the
centers of the Earth and Moon. This situation is called Tidal Lock,
and in this case, the terrestrial day, the lunar month and the Lunar
day would all be equal. At present, only the Lunar month and the Lunar
day are equal to each other, which is why the Moon presents the same
face to the Earth always.
In this picture, Lunar deformations are not commensurate in
importance to that of the Earth, because the former is much more of a
rigid body than the Earth.
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